Question

what is the inflection point of the function $f(x)=(x - 8)^3-2$?
option #1:$(8,-2)$
option #2:$(8,2)$
option #3:$(2,8)$
(1 point)
the best answer to the question is option #
check answer remaining attempts : 3

Explanation:

Step1: Find the second derivative

First, find the first derivative of \(y=(x - 8)^{3}-2\) using the chain rule. If \(u=x - 8\), then \(y = u^{3}-2\). The derivative of \(y\) with respect to \(u\) is \(\frac{dy}{du}=3u^{2}\), and the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx}=1\). So, by the chain rule \(\frac{dy}{dx}=3(x - 8)^{2}\).

Then, find the second derivative. Let \(v=(x - 8)^{2}\), and \(\frac{dy}{dx}=3v\). The derivative of \(v\) with respect to \(x\) is \(2(x - 8)\) (using the power rule \((x^{n})^\prime=nx^{n - 1}\)). So, \(\frac{d^{2}y}{dx^{2}}=6(x - 8)\).

Step2: Set the second derivative equal to zero

Set \(\frac{d^{2}y}{dx^{2}} = 0\), so \(6(x - 8)=0\). Solving for \(x\), we get \(x = 8\).

Step3: Find the \(y\) - value

Substitute \(x = 8\) into the original function \(y=(x - 8)^{3}-2\). When \(x = 8\), \(y=(8 - 8)^{3}-2=- 2\).

Answer:

A. \((8,-2)\)